How to add a new audio track to a video clip
Okay, so I'm posting this answer to my own question in case someone else has the same problem as I did. If you feel this isn't helpful, please feel free to vote to delete it. 

The issue was that I tried adding the video clip to the audio instead of the other way around. Once I reloaded the file and put in the video clip first and then deleted the original audio and added the new audio it worked just fine.